HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.4%

Time: 10.8s

Alternatives: 11

Speedup: 1.0×

Specification

\[\left(10^{-5} \leq u \land u \leq 1\right) \land \left(0 \leq v \land v \leq 109.746574\right)\]

Math

\[\begin{array}{l} \\ 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (+ 1.0 (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v))))))))

C

float code(float u, float v) {
	return 1.0f + (v * logf((u + ((1.0f - u) * expf((-2.0f / v))))));
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    code = 1.0e0 + (v * log((u + ((1.0e0 - u) * exp(((-2.0e0) / v))))))
end function

Julia

function code(u, v)
	return Float32(Float32(1.0) + Float32(v * log(Float32(u + Float32(Float32(Float32(1.0) - u) * exp(Float32(Float32(-2.0) / v)))))))
end

MATLAB

function tmp = code(u, v)
	tmp = single(1.0) + (v * log((u + ((single(1.0) - u) * exp((single(-2.0) / v))))));
end

Initial Program: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (+ 1.0 (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v))))))))

C

float code(float u, float v) {
	return 1.0f + (v * logf((u + ((1.0f - u) * expf((-2.0f / v))))));
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    code = 1.0e0 + (v * log((u + ((1.0e0 - u) * exp(((-2.0e0) / v))))))
end function

Julia

function code(u, v)
	return Float32(Float32(1.0) + Float32(v * log(Float32(u + Float32(Float32(Float32(1.0) - u) * exp(Float32(Float32(-2.0) / v)))))))
end

MATLAB

function tmp = code(u, v)
	tmp = single(1.0) + (v * log((u + ((single(1.0) - u) * exp((single(-2.0) / v))))));
end

Alternative 1: 99.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \log \left({\left(e^{\frac{-1}{v}}\right)}^{2} \cdot \left(1 - u\right) + u\right) \cdot v + 1 \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (+ (* (log (+ (* (pow (exp (/ -1.0 v)) 2.0) (- 1.0 u)) u)) v) 1.0))

C

float code(float u, float v) {
	return (logf(((powf(expf((-1.0f / v)), 2.0f) * (1.0f - u)) + u)) * v) + 1.0f;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    code = (log((((exp(((-1.0e0) / v)) ** 2.0e0) * (1.0e0 - u)) + u)) * v) + 1.0e0
end function

Julia

function code(u, v)
	return Float32(Float32(log(Float32(Float32((exp(Float32(Float32(-1.0) / v)) ^ Float32(2.0)) * Float32(Float32(1.0) - u)) + u)) * v) + Float32(1.0))
end

MATLAB

function tmp = code(u, v)
	tmp = (log((((exp((single(-1.0) / v)) ^ single(2.0)) * (single(1.0) - u)) + u)) * v) + single(1.0);
end

Derivation

1. Initial program 99.5%

   \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-exp.f32 N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \color{blue}{e^{\frac{-2}{v}}}\right) \]

   2. lift-/.f32 N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\color{blue}{\frac{-2}{v}}}\right) \]

   3. clear-num N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\color{blue}{\frac{1}{\frac{v}{-2}}}}\right) \]

   4. frac-2neg N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{1}{\color{blue}{\frac{\mathsf{neg}\left(v\right)}{\mathsf{neg}\left(-2\right)}}}}\right) \]

   5. associate-/r/ N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\color{blue}{\frac{1}{\mathsf{neg}\left(v\right)} \cdot \left(\mathsf{neg}\left(-2\right)\right)}}\right) \]

   6. exp-prod N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \color{blue}{{\left(e^{\frac{1}{\mathsf{neg}\left(v\right)}}\right)}^{\left(\mathsf{neg}\left(-2\right)\right)}}\right) \]

   7. lower-pow.f32 N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \color{blue}{{\left(e^{\frac{1}{\mathsf{neg}\left(v\right)}}\right)}^{\left(\mathsf{neg}\left(-2\right)\right)}}\right) \]

   8. lower-exp.f32 N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot {\color{blue}{\left(e^{\frac{1}{\mathsf{neg}\left(v\right)}}\right)}}^{\left(\mathsf{neg}\left(-2\right)\right)}\right) \]

   9. neg-mul-1 N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot {\left(e^{\frac{1}{\color{blue}{-1 \cdot v}}}\right)}^{\left(\mathsf{neg}\left(-2\right)\right)}\right) \]

   10. associate-/r* N/A

       \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot {\left(e^{\color{blue}{\frac{\frac{1}{-1}}{v}}}\right)}^{\left(\mathsf{neg}\left(-2\right)\right)}\right) \]

   11. metadata-eval N/A

       \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot {\left(e^{\frac{\color{blue}{-1}}{v}}\right)}^{\left(\mathsf{neg}\left(-2\right)\right)}\right) \]

   12. lower-/.f32 N/A

       \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot {\left(e^{\color{blue}{\frac{-1}{v}}}\right)}^{\left(\mathsf{neg}\left(-2\right)\right)}\right) \]

   13. metadata-eval 99.5

       \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot {\left(e^{\frac{-1}{v}}\right)}^{\color{blue}{2}}\right) \]

4. Applied rewrites 99.5%

   \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \color{blue}{{\left(e^{\frac{-1}{v}}\right)}^{2}}\right) \]

5. Final simplification 99.5%

   \[\leadsto \log \left({\left(e^{\frac{-1}{v}}\right)}^{2} \cdot \left(1 - u\right) + u\right) \cdot v + 1 \]
6. Add Preprocessing

Alternative 2: 87.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(e^{\frac{-2}{v}} \cdot \left(1 - u\right) + u\right) \cdot v \leq -1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-24, u, 8\right) \cdot u, \frac{0.16666666666666666}{v \cdot v}, \left(u \cdot u\right) \cdot \left(\frac{\frac{-2}{u} + \left(\frac{2}{v} + 2\right)}{u} - \frac{2}{v}\right)\right) + 1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* (log (+ (* (exp (/ -2.0 v)) (- 1.0 u)) u)) v) -1.0)
   (+
    (fma
     (* (fma -24.0 u 8.0) u)
     (/ 0.16666666666666666 (* v v))
     (* (* u u) (- (/ (+ (/ -2.0 u) (+ (/ 2.0 v) 2.0)) u) (/ 2.0 v))))
    1.0)
   1.0))

C

float code(float u, float v) {
	float tmp;
	if ((logf(((expf((-2.0f / v)) * (1.0f - u)) + u)) * v) <= -1.0f) {
		tmp = fmaf((fmaf(-24.0f, u, 8.0f) * u), (0.16666666666666666f / (v * v)), ((u * u) * ((((-2.0f / u) + ((2.0f / v) + 2.0f)) / u) - (2.0f / v)))) + 1.0f;
	} else {
		tmp = 1.0f;
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (Float32(log(Float32(Float32(exp(Float32(Float32(-2.0) / v)) * Float32(Float32(1.0) - u)) + u)) * v) <= Float32(-1.0))
		tmp = Float32(fma(Float32(fma(Float32(-24.0), u, Float32(8.0)) * u), Float32(Float32(0.16666666666666666) / Float32(v * v)), Float32(Float32(u * u) * Float32(Float32(Float32(Float32(Float32(-2.0) / u) + Float32(Float32(Float32(2.0) / v) + Float32(2.0))) / u) - Float32(Float32(2.0) / v)))) + Float32(1.0));
	else
		tmp = Float32(1.0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v)))))) < -1

   1. Initial program 92.6%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in v around inf

      \[\leadsto 1 + \color{blue}{\left(-2 \cdot \left(1 - u\right) + \left(\frac{1}{6} \cdot \frac{-16 \cdot {\left(1 - u\right)}^{3} + \left(-8 \cdot \left(1 - u\right) + 24 \cdot {\left(1 - u\right)}^{2}\right)}{{v}^{2}} + \frac{1}{2} \cdot \frac{-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)}{v}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 1 + \color{blue}{\left(\left(\frac{1}{6} \cdot \frac{-16 \cdot {\left(1 - u\right)}^{3} + \left(-8 \cdot \left(1 - u\right) + 24 \cdot {\left(1 - u\right)}^{2}\right)}{{v}^{2}} + \frac{1}{2} \cdot \frac{-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)}{v}\right) + -2 \cdot \left(1 - u\right)\right)} \]

      2. associate-+l+ N/A

         \[\leadsto 1 + \color{blue}{\left(\frac{1}{6} \cdot \frac{-16 \cdot {\left(1 - u\right)}^{3} + \left(-8 \cdot \left(1 - u\right) + 24 \cdot {\left(1 - u\right)}^{2}\right)}{{v}^{2}} + \left(\frac{1}{2} \cdot \frac{-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)}{v} + -2 \cdot \left(1 - u\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto 1 + \left(\color{blue}{\frac{-16 \cdot {\left(1 - u\right)}^{3} + \left(-8 \cdot \left(1 - u\right) + 24 \cdot {\left(1 - u\right)}^{2}\right)}{{v}^{2}} \cdot \frac{1}{6}} + \left(\frac{1}{2} \cdot \frac{-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)}{v} + -2 \cdot \left(1 - u\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto 1 + \left(\frac{-16 \cdot {\left(1 - u\right)}^{3} + \left(-8 \cdot \left(1 - u\right) + 24 \cdot {\left(1 - u\right)}^{2}\right)}{{v}^{2}} \cdot \frac{1}{6} + \color{blue}{\left(-2 \cdot \left(1 - u\right) + \frac{1}{2} \cdot \frac{-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)}{v}\right)}\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(\frac{-16 \cdot {\left(1 - u\right)}^{3} + \left(-8 \cdot \left(1 - u\right) + 24 \cdot {\left(1 - u\right)}^{2}\right)}{{v}^{2}}, \frac{1}{6}, -2 \cdot \left(1 - u\right) + \frac{1}{2} \cdot \frac{-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)}{v}\right)} \]

   5. Applied rewrites 5.3%

      \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(1 - u\right)}^{3}, -16, \left(1 - u\right) \cdot \mathsf{fma}\left(24, 1 - u, -8\right)\right)}{v \cdot v}, 0.16666666666666666, \mathsf{fma}\left(\frac{\left(1 - u\right) \cdot \mathsf{fma}\left(-4, 1 - u, 4\right)}{v}, 0.5, \left(1 - u\right) \cdot -2\right)\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 5.6%

      \[\leadsto 1 + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(24, 1 - u, -8\right), 1 - u, -16 \cdot {\left(1 - u\right)}^{3}\right), \color{blue}{\frac{0.16666666666666666}{v \cdot v}}, \mathsf{fma}\left(0.5 \cdot \left(1 - u\right), \frac{\mathsf{fma}\left(-4, 1 - u, 4\right)}{v}, -2 \cdot \left(1 - u\right)\right)\right) \]

   8. Taylor expanded in u around 0

      \[\leadsto 1 + \mathsf{fma}\left(u \cdot \left(8 + -24 \cdot u\right), \frac{\color{blue}{\frac{1}{6}}}{v \cdot v}, \mathsf{fma}\left(\frac{1}{2} \cdot \left(1 - u\right), \frac{\mathsf{fma}\left(-4, 1 - u, 4\right)}{v}, -2 \cdot \left(1 - u\right)\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 52.9%

       \[\leadsto 1 + \mathsf{fma}\left(\mathsf{fma}\left(-24, u, 8\right) \cdot u, \frac{\color{blue}{0.16666666666666666}}{v \cdot v}, \mathsf{fma}\left(0.5 \cdot \left(1 - u\right), \frac{\mathsf{fma}\left(-4, 1 - u, 4\right)}{v}, -2 \cdot \left(1 - u\right)\right)\right) \]

   11. Taylor expanded in u around -inf

       \[\leadsto 1 + \mathsf{fma}\left(\mathsf{fma}\left(-24, u, 8\right) \cdot u, \frac{\frac{1}{6}}{v \cdot v}, {u}^{2} \cdot \left(-1 \cdot \frac{2 \cdot \frac{1}{u} - \left(2 + 2 \cdot \frac{1}{v}\right)}{u} - 2 \cdot \frac{1}{v}\right)\right) \]

   13. Applied rewrites 59.0%

       \[\leadsto 1 + \mathsf{fma}\left(\mathsf{fma}\left(-24, u, 8\right) \cdot u, \frac{0.16666666666666666}{v \cdot v}, \left(\frac{\frac{-2}{u} + \left(\frac{2}{v} + 2\right)}{u} - \frac{2}{v}\right) \cdot \left(u \cdot u\right)\right) \]

   if -1 < (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v))))))

   1. Initial program 100.0%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in v around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 90.9%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(e^{\frac{-2}{v}} \cdot \left(1 - u\right) + u\right) \cdot v \leq -1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-24, u, 8\right) \cdot u, \frac{0.16666666666666666}{v \cdot v}, \left(u \cdot u\right) \cdot \left(\frac{\frac{-2}{u} + \left(\frac{2}{v} + 2\right)}{u} - \frac{2}{v}\right)\right) + 1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \log \left(e^{\frac{-2}{v}} \cdot \left(1 - u\right) + u\right) \cdot v + 1 \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (+ (* (log (+ (* (exp (/ -2.0 v)) (- 1.0 u)) u)) v) 1.0))

C

float code(float u, float v) {
	return (logf(((expf((-2.0f / v)) * (1.0f - u)) + u)) * v) + 1.0f;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    code = (log(((exp(((-2.0e0) / v)) * (1.0e0 - u)) + u)) * v) + 1.0e0
end function

Julia

function code(u, v)
	return Float32(Float32(log(Float32(Float32(exp(Float32(Float32(-2.0) / v)) * Float32(Float32(1.0) - u)) + u)) * v) + Float32(1.0))
end

MATLAB

function tmp = code(u, v)
	tmp = (log(((exp((single(-2.0) / v)) * (single(1.0) - u)) + u)) * v) + single(1.0);
end

Derivation

1. Initial program 99.5%

   \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
2. Add Preprocessing

3. Final simplification 99.5%

   \[\leadsto \log \left(e^{\frac{-2}{v}} \cdot \left(1 - u\right) + u\right) \cdot v + 1 \]
4. Add Preprocessing

Alternative 4: 96.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \log \left(e^{\frac{-2}{v}} + u\right) \cdot v + 1 \end{array} \]

FPCore

(FPCore (u v) :precision binary32 (+ (* (log (+ (exp (/ -2.0 v)) u)) v) 1.0))

C

float code(float u, float v) {
	return (logf((expf((-2.0f / v)) + u)) * v) + 1.0f;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    code = (log((exp(((-2.0e0) / v)) + u)) * v) + 1.0e0
end function

Julia

function code(u, v)
	return Float32(Float32(log(Float32(exp(Float32(Float32(-2.0) / v)) + u)) * v) + Float32(1.0))
end

MATLAB

function tmp = code(u, v)
	tmp = (log((exp((single(-2.0) / v)) + u)) * v) + single(1.0);
end

Derivation

1. Initial program 99.5%

   \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-exp.f32 N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \color{blue}{e^{\frac{-2}{v}}}\right) \]

   2. *-lft-identity N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\color{blue}{1 \cdot \frac{-2}{v}}}\right) \]

   3. exp-prod N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \color{blue}{{\left(e^{1}\right)}^{\left(\frac{-2}{v}\right)}}\right) \]

   4. lift-/.f32 N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot {\left(e^{1}\right)}^{\color{blue}{\left(\frac{-2}{v}\right)}}\right) \]

   5. frac-2neg N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot {\left(e^{1}\right)}^{\color{blue}{\left(\frac{\mathsf{neg}\left(-2\right)}{\mathsf{neg}\left(v\right)}\right)}}\right) \]

   6. distribute-frac-neg2 N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot {\left(e^{1}\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(-2\right)}{v}\right)\right)}}\right) \]

   7. pow-neg N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \color{blue}{\frac{1}{{\left(e^{1}\right)}^{\left(\frac{\mathsf{neg}\left(-2\right)}{v}\right)}}}\right) \]

   8. lower-/.f32 N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \color{blue}{\frac{1}{{\left(e^{1}\right)}^{\left(\frac{\mathsf{neg}\left(-2\right)}{v}\right)}}}\right) \]

   9. lower-pow.f32 N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\color{blue}{{\left(e^{1}\right)}^{\left(\frac{\mathsf{neg}\left(-2\right)}{v}\right)}}}\right) \]

   10. exp-1-e N/A

       \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{{\color{blue}{\mathsf{E}\left(\right)}}^{\left(\frac{\mathsf{neg}\left(-2\right)}{v}\right)}}\right) \]

   11. lower-E.f32 N/A

       \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{{\color{blue}{\mathsf{E}\left(\right)}}^{\left(\frac{\mathsf{neg}\left(-2\right)}{v}\right)}}\right) \]

   12. lower-/.f32 N/A

       \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{{\mathsf{E}\left(\right)}^{\color{blue}{\left(\frac{\mathsf{neg}\left(-2\right)}{v}\right)}}}\right) \]

   13. metadata-eval 99.5

       \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{{\mathsf{E}\left(\right)}^{\left(\frac{\color{blue}{2}}{v}\right)}}\right) \]

4. Applied rewrites 99.5%

   \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \color{blue}{\frac{1}{{\mathsf{E}\left(\right)}^{\left(\frac{2}{v}\right)}}}\right) \]

5. Taylor expanded in u around 0

   \[\leadsto 1 + v \cdot \log \left(u + \color{blue}{\frac{1}{e^{2 \cdot \frac{\log \mathsf{E}\left(\right)}{v}}}}\right) \]
6. Step-by-step derivation

   1. rec-exp N/A

      \[\leadsto 1 + v \cdot \log \left(u + \color{blue}{e^{\mathsf{neg}\left(2 \cdot \frac{\log \mathsf{E}\left(\right)}{v}\right)}}\right) \]

   2. log-E N/A

      \[\leadsto 1 + v \cdot \log \left(u + e^{\mathsf{neg}\left(2 \cdot \frac{\color{blue}{1}}{v}\right)}\right) \]

   3. distribute-lft-neg-in N/A

      \[\leadsto 1 + v \cdot \log \left(u + e^{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{1}{v}}}\right) \]

   4. lower-exp.f32 N/A

      \[\leadsto 1 + v \cdot \log \left(u + \color{blue}{e^{\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{1}{v}}}\right) \]

   5. distribute-lft-neg-in N/A

      \[\leadsto 1 + v \cdot \log \left(u + e^{\color{blue}{\mathsf{neg}\left(2 \cdot \frac{1}{v}\right)}}\right) \]

   6. associate-*r/ N/A

      \[\leadsto 1 + v \cdot \log \left(u + e^{\mathsf{neg}\left(\color{blue}{\frac{2 \cdot 1}{v}}\right)}\right) \]

   7. metadata-eval N/A

      \[\leadsto 1 + v \cdot \log \left(u + e^{\mathsf{neg}\left(\frac{\color{blue}{2}}{v}\right)}\right) \]

   8. distribute-neg-frac N/A

      \[\leadsto 1 + v \cdot \log \left(u + e^{\color{blue}{\frac{\mathsf{neg}\left(2\right)}{v}}}\right) \]

   9. metadata-eval N/A

      \[\leadsto 1 + v \cdot \log \left(u + e^{\frac{\color{blue}{-2}}{v}}\right) \]

   10. lower-/.f32 96.4

       \[\leadsto 1 + v \cdot \log \left(u + e^{\color{blue}{\frac{-2}{v}}}\right) \]

7. Applied rewrites 96.4%

   \[\leadsto 1 + v \cdot \log \left(u + \color{blue}{e^{\frac{-2}{v}}}\right) \]

8. Final simplification 96.4%

   \[\leadsto \log \left(e^{\frac{-2}{v}} + u\right) \cdot v + 1 \]
9. Add Preprocessing

Alternative 5: 95.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{1}{1 - \frac{-2 - \frac{\frac{1.3333333333333333}{v} + 2}{v}}{v}} \cdot \left(1 - u\right) + u\right) \cdot v + 1 \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (+
  (*
   (log
    (+
     (*
      (/ 1.0 (- 1.0 (/ (- -2.0 (/ (+ (/ 1.3333333333333333 v) 2.0) v)) v)))
      (- 1.0 u))
     u))
   v)
  1.0))

C

float code(float u, float v) {
	return (logf((((1.0f / (1.0f - ((-2.0f - (((1.3333333333333333f / v) + 2.0f) / v)) / v))) * (1.0f - u)) + u)) * v) + 1.0f;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    code = (log((((1.0e0 / (1.0e0 - (((-2.0e0) - (((1.3333333333333333e0 / v) + 2.0e0) / v)) / v))) * (1.0e0 - u)) + u)) * v) + 1.0e0
end function

Julia

function code(u, v)
	return Float32(Float32(log(Float32(Float32(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(Float32(-2.0) - Float32(Float32(Float32(Float32(1.3333333333333333) / v) + Float32(2.0)) / v)) / v))) * Float32(Float32(1.0) - u)) + u)) * v) + Float32(1.0))
end

MATLAB

function tmp = code(u, v)
	tmp = (log((((single(1.0) / (single(1.0) - ((single(-2.0) - (((single(1.3333333333333333) / v) + single(2.0)) / v)) / v))) * (single(1.0) - u)) + u)) * v) + single(1.0);
end

Derivation

1. Initial program 99.5%

   \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-exp.f32 N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \color{blue}{e^{\frac{-2}{v}}}\right) \]

   2. *-lft-identity N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\color{blue}{1 \cdot \frac{-2}{v}}}\right) \]

   3. exp-prod N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \color{blue}{{\left(e^{1}\right)}^{\left(\frac{-2}{v}\right)}}\right) \]

   4. lift-/.f32 N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot {\left(e^{1}\right)}^{\color{blue}{\left(\frac{-2}{v}\right)}}\right) \]

   5. frac-2neg N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot {\left(e^{1}\right)}^{\color{blue}{\left(\frac{\mathsf{neg}\left(-2\right)}{\mathsf{neg}\left(v\right)}\right)}}\right) \]

   6. distribute-frac-neg2 N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot {\left(e^{1}\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(-2\right)}{v}\right)\right)}}\right) \]

   7. pow-neg N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \color{blue}{\frac{1}{{\left(e^{1}\right)}^{\left(\frac{\mathsf{neg}\left(-2\right)}{v}\right)}}}\right) \]

   8. lower-/.f32 N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \color{blue}{\frac{1}{{\left(e^{1}\right)}^{\left(\frac{\mathsf{neg}\left(-2\right)}{v}\right)}}}\right) \]

   9. lower-pow.f32 N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\color{blue}{{\left(e^{1}\right)}^{\left(\frac{\mathsf{neg}\left(-2\right)}{v}\right)}}}\right) \]

   10. exp-1-e N/A

       \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{{\color{blue}{\mathsf{E}\left(\right)}}^{\left(\frac{\mathsf{neg}\left(-2\right)}{v}\right)}}\right) \]

   11. lower-E.f32 N/A

       \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{{\color{blue}{\mathsf{E}\left(\right)}}^{\left(\frac{\mathsf{neg}\left(-2\right)}{v}\right)}}\right) \]

   12. lower-/.f32 N/A

       \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{{\mathsf{E}\left(\right)}^{\color{blue}{\left(\frac{\mathsf{neg}\left(-2\right)}{v}\right)}}}\right) \]

   13. metadata-eval 99.5

       \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{{\mathsf{E}\left(\right)}^{\left(\frac{\color{blue}{2}}{v}\right)}}\right) \]

4. Applied rewrites 99.5%

   \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \color{blue}{\frac{1}{{\mathsf{E}\left(\right)}^{\left(\frac{2}{v}\right)}}}\right) \]

5. Taylor expanded in v around -inf

   \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\color{blue}{1 + -1 \cdot \frac{-2 \cdot \log \mathsf{E}\left(\right) + -1 \cdot \frac{\frac{4}{3} \cdot \frac{{\log \mathsf{E}\left(\right)}^{3}}{v} + 2 \cdot {\log \mathsf{E}\left(\right)}^{2}}{v}}{v}}}\right) \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{1 + \color{blue}{\left(\mathsf{neg}\left(\frac{-2 \cdot \log \mathsf{E}\left(\right) + -1 \cdot \frac{\frac{4}{3} \cdot \frac{{\log \mathsf{E}\left(\right)}^{3}}{v} + 2 \cdot {\log \mathsf{E}\left(\right)}^{2}}{v}}{v}\right)\right)}}\right) \]

   2. unsub-neg N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\color{blue}{1 - \frac{-2 \cdot \log \mathsf{E}\left(\right) + -1 \cdot \frac{\frac{4}{3} \cdot \frac{{\log \mathsf{E}\left(\right)}^{3}}{v} + 2 \cdot {\log \mathsf{E}\left(\right)}^{2}}{v}}{v}}}\right) \]

   3. lower--.f32 N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\color{blue}{1 - \frac{-2 \cdot \log \mathsf{E}\left(\right) + -1 \cdot \frac{\frac{4}{3} \cdot \frac{{\log \mathsf{E}\left(\right)}^{3}}{v} + 2 \cdot {\log \mathsf{E}\left(\right)}^{2}}{v}}{v}}}\right) \]

   4. lower-/.f32 N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{1 - \color{blue}{\frac{-2 \cdot \log \mathsf{E}\left(\right) + -1 \cdot \frac{\frac{4}{3} \cdot \frac{{\log \mathsf{E}\left(\right)}^{3}}{v} + 2 \cdot {\log \mathsf{E}\left(\right)}^{2}}{v}}{v}}}\right) \]

7. Applied rewrites 94.7%

   \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\color{blue}{1 - \frac{-2 - \frac{\frac{1.3333333333333333}{v} + 2}{v}}{v}}}\right) \]

8. Final simplification 94.7%

   \[\leadsto \log \left(\frac{1}{1 - \frac{-2 - \frac{\frac{1.3333333333333333}{v} + 2}{v}}{v}} \cdot \left(1 - u\right) + u\right) \cdot v + 1 \]
9. Add Preprocessing

Alternative 6: 93.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{1}{\frac{\frac{2}{v} + 2}{v} + 1} \cdot \left(-u\right) + u\right) \cdot v + 1 \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (+ (* (log (+ (* (/ 1.0 (+ (/ (+ (/ 2.0 v) 2.0) v) 1.0)) (- u)) u)) v) 1.0))

C

float code(float u, float v) {
	return (logf((((1.0f / ((((2.0f / v) + 2.0f) / v) + 1.0f)) * -u) + u)) * v) + 1.0f;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    code = (log((((1.0e0 / ((((2.0e0 / v) + 2.0e0) / v) + 1.0e0)) * -u) + u)) * v) + 1.0e0
end function

Julia

function code(u, v)
	return Float32(Float32(log(Float32(Float32(Float32(Float32(1.0) / Float32(Float32(Float32(Float32(Float32(2.0) / v) + Float32(2.0)) / v) + Float32(1.0))) * Float32(-u)) + u)) * v) + Float32(1.0))
end

MATLAB

function tmp = code(u, v)
	tmp = (log((((single(1.0) / ((((single(2.0) / v) + single(2.0)) / v) + single(1.0))) * -u) + u)) * v) + single(1.0);
end

Derivation

1. Initial program 99.5%

   \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-exp.f32 N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \color{blue}{e^{\frac{-2}{v}}}\right) \]

   2. *-lft-identity N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\color{blue}{1 \cdot \frac{-2}{v}}}\right) \]

   3. exp-prod N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \color{blue}{{\left(e^{1}\right)}^{\left(\frac{-2}{v}\right)}}\right) \]

   4. lift-/.f32 N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot {\left(e^{1}\right)}^{\color{blue}{\left(\frac{-2}{v}\right)}}\right) \]

   5. frac-2neg N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot {\left(e^{1}\right)}^{\color{blue}{\left(\frac{\mathsf{neg}\left(-2\right)}{\mathsf{neg}\left(v\right)}\right)}}\right) \]

   6. distribute-frac-neg2 N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot {\left(e^{1}\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(-2\right)}{v}\right)\right)}}\right) \]

   7. pow-neg N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \color{blue}{\frac{1}{{\left(e^{1}\right)}^{\left(\frac{\mathsf{neg}\left(-2\right)}{v}\right)}}}\right) \]

   8. lower-/.f32 N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \color{blue}{\frac{1}{{\left(e^{1}\right)}^{\left(\frac{\mathsf{neg}\left(-2\right)}{v}\right)}}}\right) \]

   9. lower-pow.f32 N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\color{blue}{{\left(e^{1}\right)}^{\left(\frac{\mathsf{neg}\left(-2\right)}{v}\right)}}}\right) \]

   10. exp-1-e N/A

       \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{{\color{blue}{\mathsf{E}\left(\right)}}^{\left(\frac{\mathsf{neg}\left(-2\right)}{v}\right)}}\right) \]

   11. lower-E.f32 N/A

       \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{{\color{blue}{\mathsf{E}\left(\right)}}^{\left(\frac{\mathsf{neg}\left(-2\right)}{v}\right)}}\right) \]

   12. lower-/.f32 N/A

       \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{{\mathsf{E}\left(\right)}^{\color{blue}{\left(\frac{\mathsf{neg}\left(-2\right)}{v}\right)}}}\right) \]

   13. metadata-eval 99.5

       \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{{\mathsf{E}\left(\right)}^{\left(\frac{\color{blue}{2}}{v}\right)}}\right) \]

4. Applied rewrites 99.5%

   \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \color{blue}{\frac{1}{{\mathsf{E}\left(\right)}^{\left(\frac{2}{v}\right)}}}\right) \]

5. Taylor expanded in v around -inf

   \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\color{blue}{1 + -1 \cdot \frac{-2 \cdot \log \mathsf{E}\left(\right) + -2 \cdot \frac{{\log \mathsf{E}\left(\right)}^{2}}{v}}{v}}}\right) \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\color{blue}{-1 \cdot \frac{-2 \cdot \log \mathsf{E}\left(\right) + -2 \cdot \frac{{\log \mathsf{E}\left(\right)}^{2}}{v}}{v} + 1}}\right) \]

   2. lower-+.f32 N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\color{blue}{-1 \cdot \frac{-2 \cdot \log \mathsf{E}\left(\right) + -2 \cdot \frac{{\log \mathsf{E}\left(\right)}^{2}}{v}}{v} + 1}}\right) \]

7. Applied rewrites 93.0%

   \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\color{blue}{\frac{\frac{2}{v} + 2}{v} + 1}}\right) \]

8. Taylor expanded in u around inf

   \[\leadsto 1 + v \cdot \log \left(u + \color{blue}{\left(-1 \cdot u\right)} \cdot \frac{1}{\frac{\frac{2}{v} + 2}{v} + 1}\right) \]
9. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto 1 + v \cdot \log \left(u + \color{blue}{\left(\mathsf{neg}\left(u\right)\right)} \cdot \frac{1}{\frac{\frac{2}{v} + 2}{v} + 1}\right) \]

   2. lower-neg.f32 93.3

      \[\leadsto 1 + v \cdot \log \left(u + \color{blue}{\left(-u\right)} \cdot \frac{1}{\frac{\frac{2}{v} + 2}{v} + 1}\right) \]

10. Applied rewrites 93.3%

    \[\leadsto 1 + v \cdot \log \left(u + \color{blue}{\left(-u\right)} \cdot \frac{1}{\frac{\frac{2}{v} + 2}{v} + 1}\right) \]

11. Final simplification 93.3%

    \[\leadsto \log \left(\frac{1}{\frac{\frac{2}{v} + 2}{v} + 1} \cdot \left(-u\right) + u\right) \cdot v + 1 \]
12. Add Preprocessing

Alternative 7: 93.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{1 - u}{\frac{\frac{2}{v} + 2}{v} + 1} + u\right) \cdot v + 1 \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (+ (* (log (+ (/ (- 1.0 u) (+ (/ (+ (/ 2.0 v) 2.0) v) 1.0)) u)) v) 1.0))

C

float code(float u, float v) {
	return (logf((((1.0f - u) / ((((2.0f / v) + 2.0f) / v) + 1.0f)) + u)) * v) + 1.0f;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    code = (log((((1.0e0 - u) / ((((2.0e0 / v) + 2.0e0) / v) + 1.0e0)) + u)) * v) + 1.0e0
end function

Julia

function code(u, v)
	return Float32(Float32(log(Float32(Float32(Float32(Float32(1.0) - u) / Float32(Float32(Float32(Float32(Float32(2.0) / v) + Float32(2.0)) / v) + Float32(1.0))) + u)) * v) + Float32(1.0))
end

MATLAB

function tmp = code(u, v)
	tmp = (log((((single(1.0) - u) / ((((single(2.0) / v) + single(2.0)) / v) + single(1.0))) + u)) * v) + single(1.0);
end

Derivation

1. Initial program 99.5%

   \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-exp.f32 N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \color{blue}{e^{\frac{-2}{v}}}\right) \]

   2. *-lft-identity N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\color{blue}{1 \cdot \frac{-2}{v}}}\right) \]

   3. exp-prod N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \color{blue}{{\left(e^{1}\right)}^{\left(\frac{-2}{v}\right)}}\right) \]

   4. lift-/.f32 N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot {\left(e^{1}\right)}^{\color{blue}{\left(\frac{-2}{v}\right)}}\right) \]

   5. frac-2neg N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot {\left(e^{1}\right)}^{\color{blue}{\left(\frac{\mathsf{neg}\left(-2\right)}{\mathsf{neg}\left(v\right)}\right)}}\right) \]

   6. distribute-frac-neg2 N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot {\left(e^{1}\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(-2\right)}{v}\right)\right)}}\right) \]

   7. pow-neg N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \color{blue}{\frac{1}{{\left(e^{1}\right)}^{\left(\frac{\mathsf{neg}\left(-2\right)}{v}\right)}}}\right) \]

   8. lower-/.f32 N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \color{blue}{\frac{1}{{\left(e^{1}\right)}^{\left(\frac{\mathsf{neg}\left(-2\right)}{v}\right)}}}\right) \]

   9. lower-pow.f32 N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\color{blue}{{\left(e^{1}\right)}^{\left(\frac{\mathsf{neg}\left(-2\right)}{v}\right)}}}\right) \]

   10. exp-1-e N/A

       \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{{\color{blue}{\mathsf{E}\left(\right)}}^{\left(\frac{\mathsf{neg}\left(-2\right)}{v}\right)}}\right) \]

   11. lower-E.f32 N/A

       \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{{\color{blue}{\mathsf{E}\left(\right)}}^{\left(\frac{\mathsf{neg}\left(-2\right)}{v}\right)}}\right) \]

   12. lower-/.f32 N/A

       \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{{\mathsf{E}\left(\right)}^{\color{blue}{\left(\frac{\mathsf{neg}\left(-2\right)}{v}\right)}}}\right) \]

   13. metadata-eval 99.5

       \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{{\mathsf{E}\left(\right)}^{\left(\frac{\color{blue}{2}}{v}\right)}}\right) \]

4. Applied rewrites 99.5%

   \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \color{blue}{\frac{1}{{\mathsf{E}\left(\right)}^{\left(\frac{2}{v}\right)}}}\right) \]

5. Taylor expanded in v around -inf

   \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\color{blue}{1 + -1 \cdot \frac{-2 \cdot \log \mathsf{E}\left(\right) + -2 \cdot \frac{{\log \mathsf{E}\left(\right)}^{2}}{v}}{v}}}\right) \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\color{blue}{-1 \cdot \frac{-2 \cdot \log \mathsf{E}\left(\right) + -2 \cdot \frac{{\log \mathsf{E}\left(\right)}^{2}}{v}}{v} + 1}}\right) \]

   2. lower-+.f32 N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\color{blue}{-1 \cdot \frac{-2 \cdot \log \mathsf{E}\left(\right) + -2 \cdot \frac{{\log \mathsf{E}\left(\right)}^{2}}{v}}{v} + 1}}\right) \]

7. Applied rewrites 93.0%

   \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\color{blue}{\frac{\frac{2}{v} + 2}{v} + 1}}\right) \]
8. Step-by-step derivation

   1. lift-+.f32 N/A

      \[\leadsto \color{blue}{1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\frac{\frac{2}{v} + 2}{v} + 1}\right)} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\frac{\frac{2}{v} + 2}{v} + 1}\right) + 1} \]

   3. lower-+.f32 93.0

      \[\leadsto \color{blue}{v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\frac{\frac{2}{v} + 2}{v} + 1}\right) + 1} \]

9. Applied rewrites 93.0%

   \[\leadsto \color{blue}{\log \left(\frac{1 - u}{\frac{\frac{2}{v} + 2}{v} + 1} + u\right) \cdot v + 1} \]
10. Add Preprocessing

Alternative 8: 91.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{1}{\frac{2}{v \cdot v} + 1} \cdot \left(1 - u\right) + u\right) \cdot v + 1 \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (+ (* (log (+ (* (/ 1.0 (+ (/ 2.0 (* v v)) 1.0)) (- 1.0 u)) u)) v) 1.0))

C

float code(float u, float v) {
	return (logf((((1.0f / ((2.0f / (v * v)) + 1.0f)) * (1.0f - u)) + u)) * v) + 1.0f;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    code = (log((((1.0e0 / ((2.0e0 / (v * v)) + 1.0e0)) * (1.0e0 - u)) + u)) * v) + 1.0e0
end function

Julia

function code(u, v)
	return Float32(Float32(log(Float32(Float32(Float32(Float32(1.0) / Float32(Float32(Float32(2.0) / Float32(v * v)) + Float32(1.0))) * Float32(Float32(1.0) - u)) + u)) * v) + Float32(1.0))
end

MATLAB

function tmp = code(u, v)
	tmp = (log((((single(1.0) / ((single(2.0) / (v * v)) + single(1.0))) * (single(1.0) - u)) + u)) * v) + single(1.0);
end

Derivation

1. Initial program 99.5%

   \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-exp.f32 N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \color{blue}{e^{\frac{-2}{v}}}\right) \]

   2. *-lft-identity N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\color{blue}{1 \cdot \frac{-2}{v}}}\right) \]

   3. exp-prod N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \color{blue}{{\left(e^{1}\right)}^{\left(\frac{-2}{v}\right)}}\right) \]

   4. lift-/.f32 N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot {\left(e^{1}\right)}^{\color{blue}{\left(\frac{-2}{v}\right)}}\right) \]

   5. frac-2neg N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot {\left(e^{1}\right)}^{\color{blue}{\left(\frac{\mathsf{neg}\left(-2\right)}{\mathsf{neg}\left(v\right)}\right)}}\right) \]

   6. distribute-frac-neg2 N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot {\left(e^{1}\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(-2\right)}{v}\right)\right)}}\right) \]

   7. pow-neg N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \color{blue}{\frac{1}{{\left(e^{1}\right)}^{\left(\frac{\mathsf{neg}\left(-2\right)}{v}\right)}}}\right) \]

   8. lower-/.f32 N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \color{blue}{\frac{1}{{\left(e^{1}\right)}^{\left(\frac{\mathsf{neg}\left(-2\right)}{v}\right)}}}\right) \]

   9. lower-pow.f32 N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\color{blue}{{\left(e^{1}\right)}^{\left(\frac{\mathsf{neg}\left(-2\right)}{v}\right)}}}\right) \]

   10. exp-1-e N/A

       \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{{\color{blue}{\mathsf{E}\left(\right)}}^{\left(\frac{\mathsf{neg}\left(-2\right)}{v}\right)}}\right) \]

   11. lower-E.f32 N/A

       \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{{\color{blue}{\mathsf{E}\left(\right)}}^{\left(\frac{\mathsf{neg}\left(-2\right)}{v}\right)}}\right) \]

   12. lower-/.f32 N/A

       \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{{\mathsf{E}\left(\right)}^{\color{blue}{\left(\frac{\mathsf{neg}\left(-2\right)}{v}\right)}}}\right) \]

   13. metadata-eval 99.5

       \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{{\mathsf{E}\left(\right)}^{\left(\frac{\color{blue}{2}}{v}\right)}}\right) \]

4. Applied rewrites 99.5%

   \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \color{blue}{\frac{1}{{\mathsf{E}\left(\right)}^{\left(\frac{2}{v}\right)}}}\right) \]

5. Taylor expanded in v around -inf

   \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\color{blue}{1 + -1 \cdot \frac{-2 \cdot \log \mathsf{E}\left(\right) + -2 \cdot \frac{{\log \mathsf{E}\left(\right)}^{2}}{v}}{v}}}\right) \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\color{blue}{-1 \cdot \frac{-2 \cdot \log \mathsf{E}\left(\right) + -2 \cdot \frac{{\log \mathsf{E}\left(\right)}^{2}}{v}}{v} + 1}}\right) \]

   2. lower-+.f32 N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\color{blue}{-1 \cdot \frac{-2 \cdot \log \mathsf{E}\left(\right) + -2 \cdot \frac{{\log \mathsf{E}\left(\right)}^{2}}{v}}{v} + 1}}\right) \]

7. Applied rewrites 93.0%

   \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\color{blue}{\frac{\frac{2}{v} + 2}{v} + 1}}\right) \]

8. Taylor expanded in v around 0

   \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\frac{2}{{v}^{2}} + 1}\right) \]
9. Step-by-step derivation

10. Applied rewrites 91.1%

    \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\frac{2}{v \cdot v} + 1}\right) \]

11. Final simplification 91.1%

    \[\leadsto \log \left(\frac{1}{\frac{2}{v \cdot v} + 1} \cdot \left(1 - u\right) + u\right) \cdot v + 1 \]
12. Add Preprocessing

Alternative 9: 91.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{1}{\frac{2}{v} + 1} \cdot \left(1 - u\right) + u\right) \cdot v + 1 \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (+ (* (log (+ (* (/ 1.0 (+ (/ 2.0 v) 1.0)) (- 1.0 u)) u)) v) 1.0))

C

float code(float u, float v) {
	return (logf((((1.0f / ((2.0f / v) + 1.0f)) * (1.0f - u)) + u)) * v) + 1.0f;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    code = (log((((1.0e0 / ((2.0e0 / v) + 1.0e0)) * (1.0e0 - u)) + u)) * v) + 1.0e0
end function

Julia

function code(u, v)
	return Float32(Float32(log(Float32(Float32(Float32(Float32(1.0) / Float32(Float32(Float32(2.0) / v) + Float32(1.0))) * Float32(Float32(1.0) - u)) + u)) * v) + Float32(1.0))
end

MATLAB

function tmp = code(u, v)
	tmp = (log((((single(1.0) / ((single(2.0) / v) + single(1.0))) * (single(1.0) - u)) + u)) * v) + single(1.0);
end

Derivation

1. Initial program 99.5%

   \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-exp.f32 N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \color{blue}{e^{\frac{-2}{v}}}\right) \]

   2. *-lft-identity N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\color{blue}{1 \cdot \frac{-2}{v}}}\right) \]

   3. exp-prod N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \color{blue}{{\left(e^{1}\right)}^{\left(\frac{-2}{v}\right)}}\right) \]

   4. lift-/.f32 N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot {\left(e^{1}\right)}^{\color{blue}{\left(\frac{-2}{v}\right)}}\right) \]

   5. frac-2neg N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot {\left(e^{1}\right)}^{\color{blue}{\left(\frac{\mathsf{neg}\left(-2\right)}{\mathsf{neg}\left(v\right)}\right)}}\right) \]

   6. distribute-frac-neg2 N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot {\left(e^{1}\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(-2\right)}{v}\right)\right)}}\right) \]

   7. pow-neg N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \color{blue}{\frac{1}{{\left(e^{1}\right)}^{\left(\frac{\mathsf{neg}\left(-2\right)}{v}\right)}}}\right) \]

   8. lower-/.f32 N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \color{blue}{\frac{1}{{\left(e^{1}\right)}^{\left(\frac{\mathsf{neg}\left(-2\right)}{v}\right)}}}\right) \]

   9. lower-pow.f32 N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\color{blue}{{\left(e^{1}\right)}^{\left(\frac{\mathsf{neg}\left(-2\right)}{v}\right)}}}\right) \]

   10. exp-1-e N/A

       \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{{\color{blue}{\mathsf{E}\left(\right)}}^{\left(\frac{\mathsf{neg}\left(-2\right)}{v}\right)}}\right) \]

   11. lower-E.f32 N/A

       \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{{\color{blue}{\mathsf{E}\left(\right)}}^{\left(\frac{\mathsf{neg}\left(-2\right)}{v}\right)}}\right) \]

   12. lower-/.f32 N/A

       \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{{\mathsf{E}\left(\right)}^{\color{blue}{\left(\frac{\mathsf{neg}\left(-2\right)}{v}\right)}}}\right) \]

   13. metadata-eval 99.5

       \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{{\mathsf{E}\left(\right)}^{\left(\frac{\color{blue}{2}}{v}\right)}}\right) \]

4. Applied rewrites 99.5%

   \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \color{blue}{\frac{1}{{\mathsf{E}\left(\right)}^{\left(\frac{2}{v}\right)}}}\right) \]

5. Taylor expanded in v around inf

   \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\color{blue}{1 + 2 \cdot \frac{\log \mathsf{E}\left(\right)}{v}}}\right) \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\color{blue}{2 \cdot \frac{\log \mathsf{E}\left(\right)}{v} + 1}}\right) \]

   2. log-E N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{2 \cdot \frac{\color{blue}{1}}{v} + 1}\right) \]

   3. lower-+.f32 N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\color{blue}{2 \cdot \frac{1}{v} + 1}}\right) \]

   4. associate-*r/ N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\color{blue}{\frac{2 \cdot 1}{v}} + 1}\right) \]

   5. metadata-eval N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\frac{\color{blue}{2}}{v} + 1}\right) \]

   6. lower-/.f32 89.9

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\color{blue}{\frac{2}{v}} + 1}\right) \]

7. Applied rewrites 89.9%

   \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\color{blue}{\frac{2}{v} + 1}}\right) \]

8. Final simplification 89.9%

   \[\leadsto \log \left(\frac{1}{\frac{2}{v} + 1} \cdot \left(1 - u\right) + u\right) \cdot v + 1 \]
9. Add Preprocessing

Alternative 10: 87.1% accurate, 231.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (u v) :precision binary32 1.0)

C

float code(float u, float v) {
	return 1.0f;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    code = 1.0e0
end function

Julia

function code(u, v)
	return Float32(1.0)
end

MATLAB

function tmp = code(u, v)
	tmp = single(1.0);
end

Derivation

1. Initial program 99.5%

   \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
2. Add Preprocessing

3. Taylor expanded in v around 0

   \[\leadsto \color{blue}{1} \]
4. Step-by-step derivation

5. Applied rewrites 85.5%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Alternative 11: 5.9% accurate, 231.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (u v) :precision binary32 -1.0)

C

float code(float u, float v) {
	return -1.0f;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    code = -1.0e0
end function

Julia

function code(u, v)
	return Float32(-1.0)
end

MATLAB

function tmp = code(u, v)
	tmp = single(-1.0);
end

Derivation

1. Initial program 99.5%

   \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
2. Add Preprocessing

3. Taylor expanded in u around 0

   \[\leadsto \color{blue}{-1} \]
4. Step-by-step derivation

5. Applied rewrites 5.8%

   \[\leadsto \color{blue}{-1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024243 
(FPCore (u v)
  :name "HairBSDF, sample_f, cosTheta"
  :precision binary32
  :pre (and (and (<= 1e-5 u) (<= u 1.0)) (and (<= 0.0 v) (<= v 109.746574)))
  (+ 1.0 (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v))))))))